You can do it on a hill. Draw one up, of your choice, and you will find that in some geometries, there are nearly straight arcs other than the equator. So you can remove portions, so long as you support the remaining intersections appropriately.

(Note that I'm not an architect, but neither was Buckminster Fuller).

Yes it is possible.

Main difference is that on a flat surface the dome weight is distributed across the base "automatically". If you just "copy-paste the default flat-terrain dome" on a sloped terrain, the bottom vertices might end up supporting more weight than the top ones. This might be ok for small/temporary domes. For something bigger/more permanent, you might want to "cut the sphere" in such a way that the weight distribution is as uniform as possible.

Also keep in mind that even on a flat surface geodesic domes a foundation - for example so that they don't fly away when there's strong wind. This would go double in sloped terrain.